Peter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

KEY ELEMENTSOF DEVELOPMENT

Part One - Directions and Mirror Structures

Peter Collins

One key element that is missing from the Integral Model (on the Integral Naked site) is - what I refer to as - the directions, which relates to the most basic manner in which dual and nondual aspects are related in experience.
Then acting as the overall operating system in development, the directions intimately affect interpretation of all other elements e.g. quadrants, levels, lines, states and types. Indeed they are more fundamental than perspectives, which are already dependent on directions. Therefore, they represent the most essential ingredient for any Integral Model that is designed to maintain proper consistency.
Each major level of development is characterised by a unique holistic mathematical manner of configuring directions (linear and circular), which then serves as the appropriate means for interpreting all other elements (at that level).
Furthermore, as refined directional appreciation depends on substantial mirror structure activity (especially at the higher levels) the discussion thereby embraces both topics.

Introduction

Looking at “What is Integral” on the Integral Naked site, we find that five key elements are mentioned as the essential ingredients of an Integral Model i.e. quadrants, levels, lines, states and types.

These five elements are indeed very important. However the most critical element is significantly missing from this model. It is - what I refer to as - the directions (or dimensions) of development. And as appreciation of these directions requires the explicit recognition of mirror (or reverse) structures that complement the standard structures of development, I will deal with both issues in these articles.

Not only is this notion of directions of fundamental importance in itself, it is vitally necessary for appropriate interpretation of all other elements of an integral approach providing the consistent interface as between dual phenomenal understanding and nondual spiritual contemplative experience of reality. [1]

I would readily admit that Ken Wilber's work is genuinely inspired by a grand integral vision. Furthermore the scope of this vision is backed up by exceptional ability allied to an amazing amount of detailed dedicated research. And I readily acknowledge that I have benefited greatly from the study of his work.

However despite this, I have never considered his cognitive interpretation of development as properly integral. Many of the problems that others have pointed to e.g. with respect to his handling of quadrants, I would see as specific symptoms of an overall method of approach that by its very nature is not designed to maintain holistic consistency. [2]

This criticism is not made lightly and I am quite confident that I can support my position. Indeed due to its fundamental importance for Integral Studies, I have consistently sought to highlight the issue since my very first appearance on a Wilber site. However by its very nature, this represents an outside criticism made from an independently formed alternative perspective. Therefore it cannot be successfully appreciated merely from within the existing Wilberian framework. Rather it requires substantial engagement with the perspective from which it is made.

The nature of linear understanding.

Most conventional discourse is characterised by linear type understanding.
The very nature of such understanding is that it is one-directional. Alternatively we could characterise it as one-dimensional. (Just as the straight line is literally one-dimensional in geometrical terms, likewise linear understanding is one-dimensional in holistic mathematical terms!). [3]
For example we tend to view the arrow of time as moving forward in a positive direction.
Likewise we think of spatial distance in the same positive manner. Thus measurements with respect to both time and space can be represented on a linear scale.

Linear understanding also typifies conventional scientific interpretation where logical asymmetric type connections are established as between variables.
So the belief that the atom is contained in the molecule (as part) rules out the opposite i.e. that the molecule is contained in the atom (as part). In other words the connection as between atom and molecule is viewed in one-directional terms.

Thus when one attempts to clarify the nature of development through making a series of unambiguous asymmetric type distinctions, then one is using the linear approach. [4]

There is of course nothing wrong per se in the use of such understanding, as it is vitally important for all phenomenal interpretation.
However properly appreciated it is strictly appropriate for mere clarification of the differentiated aspect of development.

In other words to properly distinguish phenomena in any context we must make asymmetric type distinctions.

All phenomenal understanding is conditioned by the use of polar opposites (e.g. exterior and interior) that provide a fundamental reference frame within which interpretation can take place

The importance of such polar opposites is not a new idea. This notion is deeply implicit in all of the mystical traditions and is especially significant in the formulation of Taoism.
It is also emphasised by such important Western thinkers as Heraclitus, Nicholas of Cusa, Hegel and Jung. [5]

It also plays a significant role in Ken Wilber's work with his formulation of the four quadrants (though the complementary nature of these quadrants has not been sufficiently developed).

I would see reality as being conditioned by three fundamental sets of polarities with the four quadrants accounting for two of these sets.

Thus in horizontal (heterarchical) terms, reality is conditioned by exterior and interior (and interior and exterior) aspects. This is the important subject/object distinction.

In vertical (hierarchical) terms, reality is conditioned by individual and collective (and collective and individual) aspects. This is the important whole/part distinction.

Finally in diagonal (simultaneous heterarchical and hierarchical) terms reality is conditioned most fundamentally by form and emptiness (and emptiness and form). This is the important immanent/transcendent distinction.

Now remarkably a unique holistic mathematical rationale for all these polarities exists (that quite literally is expressed as a reduced form of oneness), which when properly understood, greatly illumines their nature. Indeed in this sense development through and through is inherently mathematical with an amazingly simple basic structure.

The deeper nature of linear understanding relates to the manner in which we treat polar opposites - that necessarily condition all phenomena - in experience.

The linear approach is characterised by the clear separation of these polarities.

So conventional science is based on the belief that the world can be known objectively (in abstraction from the subjective observer). For example we have the current quest in physics to discover the ultimate laws of nature (that can be objectively verified).

Likewise conventional science is based on the separation of whole and part where the whole is treated in somewhat reduced terms as the sum of the parts. This then leads to the significant separation of quantitative from qualitative aspects of experience with the world largely interpreted in reduced quantitative terms.

Also such science is based on the separation of matter and Spirit (form and emptiness) where its phenomena are interpreted in a merely rational manner.
Indeed this rational bias is central to the traditional divide as between science and religion.

I am not questioning the great value of conventional science and indeed the immense benefits of extending the linear approach to every field of knowledge.
However by its very nature, this approach is analytical and geared to the differentiated appreciation of various phenomena. This then leads to the mass accumulation of detailed specialised understanding with respect to the various disciplines.

The basis of integration in experience is that polar opposites - that are treated as separate and independent from a linear perspective - are now understood as interdependent. In other words aspects such as exterior and interior, individual and collective and form and emptiness, are now understood as dynamically complementary and ultimately identical.

In direct terms the integral aspect is realised through immediate spiritual experience.
Thus the clearest realisation of the holistic interdependence of all phenomena comes through pure ineffable contemplation of reality.

I am sure that Ken Wilber would agree with this point! However a key problem exists in that his predominant mode of intellectual interpretation is quite inconsistent with such an integral perspective.
In other words Ken typically attempts to translate the nature of development in linear terms i.e. in attempting to affirm unambiguous asymmetric distinctions.
Thus I see his treatment of levels, states, quadrants, lines and types as riddled with inconsistency when viewed from an overall holistic perspective. This is directly due to his method of approach.

Ken attempts to use the refined linear interpretation of the centaur (vision-logic) to intellectually translate all relationships in development. However the cognitive understanding of the centaur is properly suited for interpretation of the centaur level. [6] Therefore when used as an overall cognitive approach, it necessarily distorts interpretation of all other levels (especially more advanced).

From another perspective Ken attempts to marry the states of the higher levels with the structural forms associated with the most advanced of the middle levels (i.e. vision-logic).

However increasing interpenetration of spiritual states with phenomena occurs at the higher levels, that significantly transforms the very structures of these levels. Therefore - to avoid inconsistency - an integral interpretation of development must attempt to use the transformed cognitive structures of the higher levels.

These operate in a substantially distinctive manner from Ken's use of vision-logic.

Thus as I see it, we have unique integral translations at each of the higher levels which properly reflects the increasingly integral nature of these levels.
We also have enhanced interpretations of all earlier stages from the perspective of the most advanced stage yet achieved in development.
So not alone are there several important structures that are more advanced than vision-logic but there are several increasingly enhanced versions of vision-logic (made available from the perspective of the more advanced stages).

To add greater clarity to these points I will now concentrate on the basic nature of bi-directional understanding.

Circular (Bi-directional) Understanding

We have already seen that the linear approach - which characterises the middle levels of the spectrum - is based on one-directional interpretation (where clear asymmetric distinctions are made).
So for example when one affirms - As Ken Wilber repeatedly does - the hierarchical view of development where the direction is from prepersonal to personal to transpersonal stages, then one is using linear (one-directional) understanding.

However bi-directional interpretation - which is required to maintain consistency with authentic spiritual deepening of H1 (the psychic/subtle realm) - operates in a much more refined and paradoxical manner.

In previous articles I have repeatedly relayed the example of Heraclitus: “the way up is the way down; the way down is the way up”.
Imagine one is lost in a desert and encounters a straight road!
Now to define direction in an unambiguous fashion, one could point in an arbitrary manner defining it as "up" the road. Then in terms of this reference frame the opposite direction would thereby be "down".
Thus by fixing interpretation with this arbitrary polar reference frame, we can give an unambiguous notion of direction for both "up" and "down".
However we could equally define direction with respect to the alternative reference frame.
Thus one could initially point in the other direction along the road defining it as "up", in which case the opposite direction would now be "down".
So once again by fixing interpretation with this (alternative) reference frame, we can give an unambiguous notion of direction for both "up" and "down".
However though the notion of direction can thus be defined unambiguously in an asymmetric manner with respect to each partial reference frame (considered as independent), when these polar frames are viewed in an interdependent manner, deep paradox results.
Thus what is "up" in terms of the first frame is "down" in terms of the second; likewise what is "down" in terms of the first, is "up" in terms of the second.

This example has a fundamental relevance for the nature of development, which is necessarily conditioned by polar reference frames that dynamically keep switching in experience.
So let us now attempt to clarify what Heraclitus really means in the statement "the way up is the way down; the way down is the way up".
Differentiation in experience always requires an arbitrary fixing of the reference frame with respect to these polar opposites (when interpretation in terms of the opposite frame is equally valid). For example with respect to exterior and interior polarities, typically in scientific investigation it is fixed with the exterior aspect!
However though this yields an unambiguous result, a basic linear confusion attaches to all such asymmetric understanding.
Because development moves in the same direction with respect to each frame (considered separately) we may then be tempted to believe that - in dynamic interdependent terms - it likewise moves in the same direction for both frames.
However in truth - as Heraclitus understood well - from an interactive perspective, development always moves in two opposite directions simultaneously.
And if the integral aspect is to be properly interpreted (and thereby not confused with differentiation) it must be identified with this interactive meaning.
Thus in a dynamic interactive context there are always two equally valid ways - in linear terms - of explaining any relationship.
Though each of these linear interpretations has an unambiguous relative validity (within its arbitrary context), in terms of each other, both interpretations are deeply paradoxical. Thus the movement to nondual understanding (which relates to the intuitive acceptance of such paradox) implicitly requires that one initially understand each relationship dualistically in bi-directional terms (i.e. sequentially within each reference frame as separate). Then nondual awareness derives from the intuitive recognition that both of these frames when viewed simultaneously are paradoxical in terms of each other.

Quite clearly if one is encouraged to interpret relationships unambiguously in terms of one fixed reference frame (as typically with Ken Wilber's intellectual interpretations) then it can lead to unnecessary dualistic rigidity that actually hampers nondual recognition!

Let us now go back and reinterpret the hierarchical view of development i.e. prepersonal to personal to transpersonal, from a proper bi-directional perspective!

The one-directional treatment (prepersonal to personal to transpersonal) is rooted in an unduly transcendent view of development. [7]
However there are two spiritual directions i.e. immanent and transcendent. The transcendent entails going beyond to find Spirit literally without phenomena (as emptiness). However by contrast immanence requires going inside form to find Sprit within (as revealed through phenomena).
We can associate growth in development with either its transcendent or immanent direction.
Thus from one perspective development can be seen as the gradual refinement in the experience of transcendence, whereby one more easily goes beyond limited phenomenal expressions of form to discover Spirit as true emptiness.
However from an equally valid perspective, development can be see as the gradual refinement in the experience of immanence, whereby one can more easily go within phenomena to discover them as transparent mediators of spiritual light.

Thus to associate the forward direction with transcendence (which Ken Wilber repeatedly does in his writings) is clearly unbalanced and arises from a linear method of approach.

In earliest development - before proper differentiation of structures takes place - both the transcendent and immanent directions are necessarily confused with each other.
The infant cannot properly transcend form (because it is still entangled with emptiness). Likewise the infant cannot properly discover Spirit as immanent in form (because emptiness is likewise entangled with form).
Put another way, both prepersonal and transpersonal aspects are necessarily confused as development commences.
Thus to refer to the earliest stages as prepersonal is inaccurate from a dynamic perspective (as this requires a level of differentiation from transpersonal which has not yet taken place). Thus it is more correct to refer to earliest development as both prepersonal and transpersonal (and transpersonal and prepersonal) in a confused manner. 8 However as development unfolds through the earlier stages the differentiation of pre and trans (and trans and pre) gradually takes place. Then by the middle stages - where differentiated interpretation becomes specialised - prepersonal is largely differentiated from trans. In other words we now have the unfolding of personal development where confused pre and trans elements largely disappear.

Then with the unfolding of the higher levels, pre and trans (and trans and pre) once more dynamically interact with each other (this time in a largely mature manner). Then with pure specialised contemplative awareness (at the highest nondual stage) pre and trans (and trans and pre) are identical in pure realisation of Spirit.

Again it is quite inaccurate to refer to the highest stages of development in discrete asymmetrical terms as transpersonal, when the very essence of such stages is the dynamic manner in which they smoothly interact - and are thereby integrated - with corresponding lower stages.
So once again the discrete interpretation of stages - where they are clearly separated from each other - is suited for the merely differentiated appreciation of development. However integral appreciation more properly relates to the continuous interpretation where both higher and lower (and lower and higher) stages are bi-directionally interdependent.

So once more when we attempt to interpret the continuous integral appreciation of stage development in a reduced linear (asymmetrical) manner, two equally valid ways are available.

Thus from one perspective we can look at development - as is typical with Ken Wilber - from the transcendent direction as the progressive growth in Spirit (as emptiness). Here we have the development from (confused) body to mind to (mature) Spirit.

However from an equally valid perspective we can look at development from the immanent direction as the progressive growth in Spirit as refined form (which then can more transparently mediate the divine light).
Here the development is from confused Spirit (i.e. where initially it is totally confused with bodily form) to mind to mature body (that now, largely free of shadow interference, can adequately reflect the spiritual light).

So once again any attempt to unambiguously portray development hierarchically e.g. as from body to mind to Spirit is very misleading from an integral perspective.

Now, within their respective frames of reference both linear interpretations of development (transcendent and immanent) appear unambiguous.
However when we simultaneously attempt to embrace both frames, deep paradox results.
Thus what is forward in terms of the transcendent direction is backward in terms of the immanent. Likewise what is forward in terms of the immanent is backwards in terms of the transcendent.
Put another way what is pre (from the transcendent perspective) is trans (from the immanent); likewise what is trans (from the immanent perspective) is pre (from the transcendent).

Thus from the appropriate integral perspective the distinction in development is not so much as between pre and trans (which reflects differentiated interpretation) but rather as between confused notions of the relationship as between pre and trans (and trans and pre) at the lower and corresponding mature notions at the higher stages.

Ultimately of course both differentiated and integral aspects are brought together in mature manner in - what I refer to as - the radial stages.

Now though pre and trans are united in absolute terms as pure spiritual awareness, which is present in all activities, in any arbitrary phenomenal context pre can be distinguished from trans (and trans and pre).
However it is important to bear in mind that such relative distinctions take place with respect to both immanent and transcendent directions (as discretely defined) thus facilitating their continual integration as pure spiritual awareness.

When we approach development from an appropriate bi-directional standpoint (which befits an integral approach) the very nature of interpretation with respect to every development issue changes substantially.
So for example, the pre/trans fallacy in the manner formulated by Ken Wilber would be impossible to sustain from an appropriate integral perspective.

I will give just one more example here from his writings.

Ken regularly identifies scientific phenomena (which he misleadingly refers to as “its”) with his Right Hand quadrants.

However from the appropriate integral perspective (i.e. bi-directional) ambiguous identification of phenomena with any one quadrant is untenable. [9]

Let us now look more closely at this issue to identify the problem.

The actual experience of scientific phenomena e.g. rocks, requires the dynamic interaction of interior mental constructs (perceptions and concepts) with objective phenomena that are - in relative terms - viewed in exterior terms.

Thus the actual experience of “a rock” as an object in the exterior environment is indistinguishable from the corresponding perception of “a rock” which - relatively - is interior.
Thus in dynamic terms the arbitrarily defined “Right Hand” quadrant of exterior phenomena coincides with the “Left Hand” quadrant of interior mental constructs.

Thus when we attempt to render this dynamic bi-directional relationship in linear (asymmetrical) terms, two equally valid interpretations are possible.

We can view scientific phenomena as independent exterior objects (existing in the Right Hand quadrants).

Alternatively from an equally valid perspective, we can view these phenomena as independent interior constructs (existing - relatively - in the Left-Hand quadrants).

Now in terms of conventional science, it does not really matter which interpretation is used (as the dynamic interaction as between both is formally understood as neutral).

In other words in such science the double correspondence view is accepted.

Phenomenal objects accurately correspond with mental constructs (and are not altered through such interaction).
Equally - from the other perspective - mental constructs correspond with related phenomenal objects (and are likewise not altered through such interaction). [10]
Of course though these assumptions are not strictly valid, conventional science works because it yields acceptably accurate approximations (based on such assumptions).

However dynamically speaking this paradigm of science is very limited and quite unsuited for holistic integral interpretation of reality.

So even in linear terms, science does not (exclusively) belong to Right-Hand quadrants, which requires the untenable assumption that an empirical object somehow has meaning in itself.
Science equally belongs to the Left-Hand quadrants i.e. in the mental constructs that are used to interpret phenomenal data. [11]

From an integral perspective, through interaction as between objects and constructs (and constructs and objects) dynamic transformation takes place. Where this is limited, such transformation takes place within the broadly accepted conventional paradigm of science.

However when the interaction increases to a sufficient level (as with the more advanced stages of spiritual development) the overall paradigm itself changes.

In other words we then properly have the birth of integral (holistic) science that is explicitly based on bi-directional - rather than linear - interpretation of relationships.

And just as conventional science is rooted in analytic mathematics (as its indispensable tool) integral science is rooted in holistic (i.e. integral) mathematics as its own indispensable tool.
So as scientific understanding of the integral nature of development requires corresponding holistic interpretation of mathematical relationships, my own integral scientific approach is intimately related to holistic mathematical translation of all key development issues. [12]

Basic Nature of Holistic Mathematical Approach

The conventional notion of number - which is synonymous with quantitative notions of order - is literally defined in a one-dimensional manner.

Foe example if we take the well-known natural numbers 1, 2, 3 and 4, implicitly they are all defined with respect to the 1st dimension and can be represented geometrically by a line (i.e. that is one-dimensional) divided into equal unit lengths.

If however we take a number that is initially expressed to another power its ultimate quantitative value is again expressed with respect to the first dimension.

Thus the value of 2^2 (i.e. 2 to the power of 2) is 4 when expressed with respect to the default 1st dimension (i.e. 4 raised to the power of 1). [13]

Clearly the linear notion of number is extremely important in scientific analytical terms as a quantitative means of ordering data.

However there is an equally important circular interpretation of number - that is as yet little appreciated - that has remarkable holistic capacity to scientifically order qualitative data. Thus for example - when appropriately understood in scientific integral terms - all stages of development can be best encoded using this circular notion of number.

The clue to the nature of this alternative type of number first dawned on me through the study of Hegel (West) and Taoism (East). So if for example we take Hegel's well known triadic arrangement, commonly represented as thesis, antitheses and synthesis, we can view the original thesis as what is posited in experience, then the antithesis as its negation, with the new synthesis arising from the creative fusion of what is now both posited and negated.
So we have here the holistic dynamic equivalents (in positing and negation) to the well-known mathematical operations of addition and subtraction. [14]
So just as 1 – 1 = 0 in analytic mathematical terms, ultimately what is posited (as unitary form in experience) and then freely and fully negated (with respect to possessive identification) results in nothingness as the experience of spiritual emptiness.

Thus 1 – 1 = 0 has an important analytic meaning in quantitative scientific terms. Equally it has an utterly profound holistic meaning from a qualitative spiritual perspective.

Then a particular expression of Taoism i.e. that the indivisible primal union splits at a phenomenal level into the opposite poles of yin and yang, led me to another key insight which is associated with the mathematical notion of a root.

If we take the simplest case of 1 and attempt to extract its square root we obtain two equally valid opposite answers i.e. + 1 and – 1.
What in effect we are doing here is interpreting a higher order two-dimensional expression in (reduced) one-dimensional terms (i.e. if x^2 = 1 then x = + 1 or – 1).

Now the bi-directional approach I have been illustrating is the extremely important holistic equivalent of this relationship.

In other words when we attempt to express the nature of integral unitary form (where opposite polarities of experience are identical) in reduced linear terms, then two equally valid interpretations are possible (which are the opposite of each other). [15]

So the vital implication of this can be briefly stated as follows.
When we express, what is literally the higher two dimensional reality of H1, in the reduced one-directional analytic terms of the middle levels we always obtain two equally opposite interpretations for any event in development.
Quite literally whereas at H0 (vision-logic) we still tended to translate developmental issues in unambiguous linear terms (where one explanation is offered) from the perspective of H1 (subtle) we now clearly see that for every interpretation we can give for any issue, there exists an opposite alternative interpretation (that is equally valid). Though both of these interpretations are linear (and unambiguous) within their respective frames of reference, when considered simultaneously in relation to each other they are deeply paradoxical. Indeed such paradoxical recognition (at an intellectual level) is the catalyst for purely spiritual awareness (where the paradox is intuitively reconciled) and constitutes the basis of the integral approach.

The implication of this integral approach (when grasped) is truly revolutionary for it requires us to look at every issue in development in an utterly distinctive manner. [16]

The two roots of unity can be expressed in geometrical terms as the circle (of unit radius) that is bisected by its line diameter (in opposite directions from the centre point).
In other words corresponding to the two roots is a circular notion of number.

If we obtain the four roots of unity we now geometrically obtain what literally corresponds to the four quadrants of the circle. (By extension if we take - say - the 100 roots of unity they will be geometrically depicted as 100 equidistant points on the circle of unit radius. So a well-defined circular notion of number is associated with such roots). [17]

The four quadrants are of immense significance in holistic mathematical terms.

In analytic terms the four roots of unity are + 1, -1 + i and – i (where i is the square root of – 1).

However though convinced - since the early 70's - of the holistic mathematical importance of these four quadrants for development, it took me about 10 years to properly interpret the holistic significance of imaginary numbers.
Indeed it was the reading of Jung - especially his treatment of Personality Types - that proved immensely helpful in this regard.
As is widely known, Jung based his personality types on four functions i.e. sense, intuition, thinking and feeling. Sense and intuition are opposites as are thinking and feeling.

Thus when once function is consciously posited in experience its opposite is manifested in an unconscious manner. Typically in personality one function will tend to dominate. Then the opposite (unconscious) function appears as projection (with its true nature misunderstood).

So the conscious (dominant) function is real and the unconscious (inferior) function literally imaginary (as the indirect expression of the unconscious). [18] Thus in holistic mathematical terms, the interplay of conscious and unconscious in psychological experience entails the continual interaction of both real and imaginary notions of form.

Thus when we scientifically allow for the interaction of conscious and unconscious reality, then - by definition - it necessarily becomes complex i.e. with both real and imaginary aspects. [19]

This has far-reaching implications for development.

For example it requires fundamental reformulation of the very notion of a holon.

A holon is customarily defined as a whole/part where a whole is also part of another whole. This definition corresponds to real (conscious) interpretation where for example the atom is rationally understood as part of a molecule.
However a holon (or rather onhol) is also a part/whole where a part is understood as whole (in the context) of other parts.
However this latter recognition, whereby distinct phenomena can be properly viewed as unique wholes, comes from the imaginary unconscious that arises intuitively in experience.
Not surprisingly therefore, conventional science - based solely on real rational interpretation - leads to considerable fragmentation of phenomena, where holons lose their individual uniqueness.

However from the “higher” integral perspective of H2 (i.e. causal realm) not alone are all relationships considered in proper bi-directional terms but equally are given both real and imaginary interpretations.

As we have seen Ken Wilber in his treatment of holonic quadrants tends to fragment their nature (through lack of appropriate bi-directional interpretation).

However his interpretation is equally reduced due to failure to properly recognise the complex nature of the holon, where it continually switches as between its real (as part of another whole) and imaginary status (as whole in the context of other parts).

So for example we cannot abstract heaps from holons. [20] Once the complex nature of holons is grasped, it becomes apparent that holons cannot exist in the absence of heaps. So Ken's treatment of heaps is very fragmented resulting from an inadequate method of interpretation (that is not sufficiently refined in bi-directional terms). [21]

One other extremely important implication of this approach is the recognition that the integral interpretation of the dimensions of space and time correspond exactly (in holistic terms) with the mathematical interpretation of dimensions.
So once again the structure of reality is mathematical in this holistic sense. [22]

Thus as I define it, integral approaches are bi-directional.
In the Integral 1 approach - which is consistent with the refined intellectual appreciation of H1 (subtle realm) - all relationships are understood in bi-directional real terms (pertaining to conscious appreciation).

With the Integral 2 approach - which in turn is consistent with the even more refined cognitive appreciation of H2 (the causal realm), all relationships are defined in bi-directional terms, with respect to both real (conscious) and imaginary (unconscious) aspects of development.
The most advanced of the strictly integral approaches (Integral 3) is consistent with the understanding of H3 (approaching pure nondual awareness). This entails bi-directional understanding with respect to both real (conscious), imaginary (unconscious) and both real and imaginary (simultaneous conscious and unconscious) aspects.

It is based on the holistic mathematical interpretation of the eight roots of unity, which in geometrical terms are represented by the eight sectors (i.e. a circle of unit radius divided into eight equal parts by horizontal, vertical and diagonal lines).

Now the remarkable feature of these additional diagonal lines (that are equidistant from real and imaginary lines) is that they have a magnitude = 0.
In physics these are known as null lines and accurately describe the nature of (physical) light.

However the holistic nature of the diagonal lines is of immense significance basically interpreting the nature of spiritual light (where form and emptiness coincide).
In other words when dynamic balance in personality is maintained as between both the heterarchical and hierarchical aspects of form with respect to both conscious and unconscious understanding, the purest level of spiritual awareness (as emptiness) is experienced. In other words true integration as the interdependence of all reality results.

Radial development then entails the mature transformation of reality as form through growing interpenetration with pure contemplative spiritual awareness.
Once again I would distinguish three levels Radial 1, Radial 2 and Radial 3.
Radial 1 entails the gradual restoration of mature transparent phenomena with respect to the diagonal polarities (form and emptiness). So it entails experience of the world that is still very close to contemplative awareness.

Radial 2 entails the additional restoration of phenomena with respect to vertical polarities where local activities (as part) now serve as a better expression of an overall global contemplative vision.

Radial 3 entails the restoration likewise of horizontal polarities (interior and exterior) in the most committed active expression of this vision.

Configuring the Directions

With this holistic mathematical interpretation of directions we can successfully configure all of the major levels of development (12 in my approach). See Part 2 for more details!

Basically the three middle levels are defined in linear (one-directional) terms with respect to merely conscious understanding.
L0 (concrete rational) is largely confined to more superficial empirical type relationships. Thus in general terms it is still rooted in bi-directional mythic confusion.
L0, H0 (formal rational) - which is the most purely rational stage - can now deal with the most abstract concepts in a linear manner.
H0 (vision-logic) leads to much more varied analytic ability that is considerably infused by the spiritual understanding pertaining to higher stages. However explicitly its structure of understanding is still one-directional.

The first of the higher levels H1 (psychic/subtle) is then defined with respect to two directions. So H1 is bi-directional in real (conscious) terms.
In complementary lower level manner L1 (i.e. mythic) is likewise defined as bi-directional in the same manner. However, whereas at the higher level the two directions are developed in mature fashion (where they are properly differentiated from each other), at its lower counterpart the two directions are still somewhat confused with each other (due to lack of sufficient differentiation).

The second of the higher levels H2 (causal) is structurally defined with respect to four directions. (So the causal level in this sense is literally four-dimensional in holistic mathematical terms)! So now all relationships are understood as bi-directional in both real (conscious) and imaginary (unconscious) terms.
This then provides the appropriate holistic mathematical means of consistently interpreting the four quadrants in a dynamic experiential manner.
Again in complementary fashion, the structure of L2 (magical) is similar. However once again, whereas at the higher level the directions are understood in mature manner (with respect to both real and imaginary aspects) at the corresponding lower stage these same directions are confused. So at L2 we have the confusion of real (conscious) with imaginary (unconscious) modes of understanding.

The third of the higher levels H3 (nondual) is defined with respect to eight directions. This level is literally eight-dimensional in holistic mathematical terms, with remarkably the four complex dimensions eventually approximating to empty nondual experience as nothingness. This likewise provides the appropriate means of consistently interpreting the eight sectors in dynamic experiential terms leading to a holistic mathematical TOE (applicable to all development relationships).
Now all relationships are understood in bi-directional manner with respect to real (conscious), imaginary (unconscious) and complex (directly psychophysical) phenomena.
In complementary lower level terms, L3 has the same bi-directional structure except that whereas all directions are ultimately understood in a mature manner at the higher level H3, they are ultimately understood in a totally confused fashion at L3.
So at the beginning of life the infant foetus cannot distinguish the most basic (physical) phenomena of form from (psychological) emptiness.

Coming from the attainment of specialised contemplative awareness, R1 (Spiritual Reawakening) relates to growing interpenetration of the three bi-directional stages (integrating higher and lower levels) with the diagonal poles of one-directional understanding (i.e. distinction of form and emptiness). In intellectual terms, this would be associated with a very generalised - though highly refined - form of enhanced intuitively based vision-logic understanding.

R2 (Mature Vision) relates to additional interpenetration of the three-bi-directional levels with the vertical poles (distinction of whole and part)
In intellectual terms this would be associated with a growing ability to embrace various disciplines in a more detailed analytic - yet still highly creative - fashion.

R3 (Creative Transformation) relates finally to the additional incorporation in full measure of the horizontal poles (distinction of interior and exterior).
This now leads to the much more practical ability to engage with respect to the world in an extremely committed and intensive fashion.
In intellectual terms this would be associated with the more active missionary process of successfully bringing the “good news” (of truly valuable ideas) to as wide an audience as possible (which requires personal as well impersonal abilities).

Notes

1. I frequently refer to an approach that is properly designed to deal with the interactive nature of reality as dynamic.
Though Ken Wilber offers a superb outline of development (within his own frame of reference) I would consider it - in terms of structure - as a mechanical rather than dynamic account.
However, though far less comprehensive that Wilber in many ways, I would acknowledge that many other writers do inherently adopt a dynamic approach to development. So for example Hegel (philosophy), Jung (psychology) and Underhill (mysticism) would readily fall into this category.

2. There is a considerable discontinuity in the manner in which Ken Wilber typically treats development, which relates in turn to a marked separation of states and structures. Up to and including the middle stages, Ken's approach is decidedly Western with its characteristic emphasis on analytic structure.
However in his treatment of higher stages his approach is decidedly more Eastern with undue emphasis on the spiritual states (largely to the exclusion of meaningful structures) of these levels. How else for example can one explain Ken's failure to properly incorporate the bi-directional structures of form of the higher levels in his integral approach?
Ken in fact never really gets to grip with the important manner in which states interact with structures (and structures with states) throughout development.
The true marriage of East and West is not achieved by attempting to combine higher level states with lower and middle level structures in a somewhat disconnected manner. Rather it comes from recognising that states and structures are necessarily in dynamic interaction throughout development.

3. A line can indeed be drawn to show two directions. So starting from a given point (0), if we draw the line horizontally in the right direction it will indicate positive and if drawn to the left negative values respectively. However changing direction requires switching the frame of reference. In other words, we can draw the line either to the right or left of the starting point but not in both directions simultaneously.
So once we decide on the frame of reference the line is necessarily one-directional. In like manner once we decide on the polar frame of reference (in any given context) linear understanding is likewise necessarily one-directional.

4. The use of a linear approach does not of course imply the belief that the nature of overall development is linear. One could for example portray development - as Ken Wilber frequently does - as an assemblage of different lines and states that can reach very different levels. However this leaves for integral explanation the overall way that these various lines and states interact with each other. And apart from very vague general statements e.g. that the self attempts to integrate the various lines, levels and states, Ken offers no satisfactory holistic explanation of the nature of such interaction.

5. F. David Peat in his “Blackfoot Physics” indicates the importance of the number 4 - representing the Four Winds and Four Directions - to the indigenous natives of Turtle Island. It was embodied in symbols such as the sacred hoop and medicine wheel representing on the one hand a state of balance and harmony and on the other a dynamic movement of spiritual forces within the cyclical nature of time. So they maintained in fact a dynamic interactive notion of the four quadrants.

Now one could perhaps maintain - from a Wilberian perspective - that this indigenous knowledge was of a prepersonal rather than transpersonal kind. However this would miss the point for in dynamic terms prepersonal and transpersonal necessarily interact with each other (so that we cannot have one aspect without the other).

6. As one moves to more advanced levels, the perspective on earlier stages continually changes leading to an increasingly enhanced appreciation of these stages.
So with respect to vision-logic we have the standard (default) interpretation, which unfolds at the centaur stage (vision-logic 1). However we have increasingly enhanced versions from the perspective of H1, H2, H3, R1, R2 and R3 which we could refer to as vision-logic 2, vision-logic 3, vision-logic 4, vision-logic 5, vision-logic 6 and vision-logic 7 respectively. Thus the most enhanced version of vision-logic available is the perspective afforded by the R3 stage.
We could also perhaps refer to the diminished versions of this same structure, which relate to the reduced interpretations afforded by lower stages. So we could perhaps refer to the diminished vision-logic interpretation of the previous formal stage L0, H0 as vision-logic - 1. However in general, diminished versions are less meaningful in experience than their enhanced counterparts.
Then as well as the discrete linear interpretation of both enhanced and diminished stages, we could also have an alternative classification based on circular complementary connections.
So for example - in this context - H0 (vision-logic) represents the enhanced version of L0 (mythic). In other words H0 (with a general spiritual vision coming from the higher levels) complements M0 (with a general mythic vision deriving from the lower levels).

7. Again this precisely represents one-directional understanding as I define it. There are two directions to development (either of which can be taken in a forward manner). So to identify - as Ken Wilber does - the forward direction of development exclusively with transcendence is clearly linear interpretation.

8. Put another way, before differentiation of any distinct stages in development takes place, the infant has access to all stages (which are necessarily confused with each other). So transpersonal are necessarily confused with prepersonal (and prepersonal with transpersonal). Though a very limited degree of personal development must also be present (once differentiation of structures commences) a much clearer link exists as between complementary extremes (in confused manner) in earliest development.

9. I am fully aware that Ken maintains that ultimately all quadrants are interdependent in integral terms. However the manner in which he then attempts to differentiate his quadrants in an unambiguous fashion (e.g. all “its” belong to the Right-Hand quadrants) is clearly inconsistent with the integral perspective. In other words Ken offers no satisfactory manner of reconciling the differentiated with the integral view (which is the central problem generally with respect to his overall approach).

10. Though in formal terms this is the position in conventional science, clearly from a dynamic perspective it is unsatisfactory. Thus through the process of doing science i.e. applying scientific hypotheses to data (and vice versa), the nature of understanding changes, leading eventually to the formulation of revolutionary new hypotheses which can be applied to a wider range of empirical data.

Also in quantum mechanics for example it is not possible to maintain the neutrality of the observer from what is observed. However despite this, physicists still seek objective type explanations (though of a probabilistic nature) within quantum mechanics.

Also the phenomena of parascience - though they may be readily dismissed in conventional terms - cannot be properly explained without allowing for the interaction of both physical and psychological aspects of experience.

11. M-theory - which is the main candidate in physics for a TOE - is a largely mathematical explanation of reality. So in Wilberian terms this theory of “its” would belong to the Left-Hand (and not Right- Hand) quadrants. So once again it makes little sense to try and locate activities of any kind unambiguously in particular quadrants.

12. Though I welcome Ken Wilber's integral mathematics of perspectives as a promising development, I would see it as largely representing a convenient notation for the expression of distinct perspectives. It is not integral therefore in the sense of providing a holistic mathematical means of interpreting the overall nature of development.

13. There is an important - largely unacknowledged - problem here with conventional mathematical interpretation.

In reduced quantitative terms 2^2 (i.e. 2 raised to the power of 2) = 4 (i.e. 4 raised to the power of 1) which can be shown geometrically by marking off 4 units on the straight-line interval (representing numbers).
However to accurately represent 2^2 in geometrical terms we need to draw a square (with unit side 2 units). So, strictly speaking 2^2 = 4 square - rather than 4 linear - units.
Thus when we raise a number to any power a qualitative - as well as quantitative - change in the nature of the units is involved. This is likewise the case with the inverse operation of taking roots. However conventional mathematical interpretation - because of its philosophical linear approach - can only offer a reduced - merely quantitative - interpretation of such numerical transformations.

14. Dynamic positing and negation (the holistic equivalent of the mathematical operations of addition and subtraction) can be extended to every perception and concept where polar opposites are involved. And as all experience is conditioned by the three fundamental polarities (interior/exterior, whole/part and form/emptiness) this means that such holistic mathematical operations apply to all phenomena.

15. The nature of holistic interpretation of polar opposites (represented by the circular diagram) is however inherently more dynamic than in analytic terms.

In conventional mathematics the positive real axis is drawn horizontally to the right (from a central point) and the negative to the left. However this is a merely arbitrary convention. It would be equally valid to standardise interpretation where all positive numbers are shown on the left axis and all negative on the right.
So in other words it is important from the analytic perspective is achieve an unambiguous standardised approach with respect to such conventions.
However in dynamic terms, such polar locations keep switching in experience (especially when keenly reflected in the light of nondual spiritual awareness).
This means with respect to quadrant interpretation, that the location of phenomena keeps changing (as polar reference frames likewise keep changing) in experience.
For example we can initially arbitrarily designate scientific “its” with either the Right-Hand or Left-Hand quadrants. However as polar reference frames continually switch through the interaction of exterior and interior (and interior and exterior) aspects of experience, these quadrant locations will themselves keep switching.
So Ken Wilber's attempt to unambiguously identify phenomena with particular quadrants (e.g. “its” with the Right-Hand) arises from an unsatisfactory - merely analytic - way of dealing with dynamic interactive experience.

16. What this implies is that a unique interpretation of development (with respect to all its concepts) arises from the application of the cognitive structures associated with the corresponding level of development.
Therefore if we identify mature interpretations with the advanced middle level (vision-logic) and higher, then we have unique interpretations for H0 (vision-logic), H1 (psychic/subtle), H2 (causal), H3 (nondual), R1 (Radial 1), R2 (Radial 2) and R3 (Radial 3).

As I have stated the cognitive approach of H0 is linear (one-directional).
The cognitive approach of the three higher levels is then bi-directional. At H1 bi-directionality is confined to the interpretation of real phenomena (that are consciously understood). At H2, bi-directionality now extends to both real (conscious) and imaginary (unconscious) interpretation of variables. At H3 this bi-directional approach is extended to include interpretation of the simultaneous (i.e. direct psychophysical) nature of phenomena in experience.
The three radial approaches then incorporate mature linear appreciation (in arbitrary relative contexts) with bi-directional holistic understanding.
With R1 the bi-directional approach is incorporated with the most general linear understanding of a fixed nature (i.e. overall structures of form).
At R2 more detailed linear structures (of whole and part systems) can be incorporated. At R3 the most active involvement in the world (from a holistic contemplative perspective) is possible through incorporation of remaining interior and exterior aspects (at a relative dualistic level).

My own particular concern has not been in providing a detailed account of development with respect to any of these levels but rather in clarifying the nature of the holistic structures of all levels (which then dictates precisely the overall nature of the interpretations involved). This is in itself a massive task, the importance of which has scarcely yet even been recognised.
In other words, present attention in Integral Studies is largely focussed on the detailed mapping of various vision-logic type interpretations. However this is taking place without sufficient regard for the inevitable holistic inconsistency of such interpretations and the corresponding need for more structurally advanced models that can remedy these deficiencies.

17. Such a circle must however be drawn in the complex plane, In other words, starting from the centre of the circle the horizontal line to the right (i.e. axis) represents real numbers that are positive and the corresponding line to the left real numbers that are negative.
Then again staring from the centre the vertical line upwards (at right angles to horizontal axis) represents imaginary numbers that are positive.
The corresponding direction of the line downwards represents imaginary numbers that are negative.

18. I would consider the holistic mathematical interpretation of the imaginary number i (the square root of –1) as the single most important breakthrough in my holistic mathematical understanding.
It has to be remembered however that ultimately every mathematical symbol and relationship in conventional terms (with a well-defined analytic meaning) has a corresponding holistic interpretation with direct relevance for the scientific integral interpretation of development. So we really are at the very beginning in terms of appreciation of this radical new interpretation of mathematics. In general I confidently predict that relationships that are already considered of fundamental importance in analytic terms, will eventually be seen to have an equally important - though often most unexpected - holistic relevance for integral interpretation.

So let us now explain more carefully the holistic mathematical interpretation of i (i.e. the square root of – 1).
In analytic terms we could express i as the value of x for x^2 = – 1.
In other words i in fact reflects a reduced one-dimensional interpretation of – 1 (that is initially defined in two-dimensional terms).

Now in holistic terms what is posited relates to conscious experience. This results in the separation of polarities leading to linear (i.e. one-dimensional) interpretation.
However if what has been posited is then negated this directly relates to the unconscious.
So the negative direction of experience relates directly to the unconscious aspect. However the very nature of the unconscious is that it is two-dimensional (where opposite polarities are combined).
Therefore if we then attempt to indirectly express what is unconscious (and two-dimensional) in one-dimensional linear terms, we are dealing with the holistic equivalent of what is imaginary (from a mathematical perspective).
So for example all projections relate to imaginary rather than real conscious experience. In other words they arise as the indirect expression of the unconscious.
By their very nature such projections are generally confused (where they become embedded in consciously understood symbols).

So once we reach a certain level of recognition (associated with causal understanding) all development is understood as complex (i.e. with both real and imaginary aspects). Therefore adequate interpretation of development at this level must likewise be complex (showing the interaction of real and imaginary aspects of experience).

19. One important illustration of the relevance of the imaginary aspect in the physical realm is provided by virtual particles. Virtual particles lie very close to the fundamental ground of nature (which is the physical expression of the unconscious) and are freely emitted from that ground.
As the inherent nature of this ground is that opposite polar aspects are inseparable, virtual particles can enjoy but a very fleeting existence, where one pole (as particle) becomes momentarily separated from the other (as anti-particle). However the closer these particles lie to the fundamental ground the more short-lived must their phenomenal existence necessarily be.

It is very similar likewise in mature psychological terms at the high causal level of development. Because they now are very close to the psychological ground (i.e. the pure spiritualised unconscious), phenomena that are emitted from the ground can only enjoy a very fleeting existence as separate phenomena (where polarities are momentarily separated). Therefore as one reaches a pure (unconscious) spiritual state, phenomena are so short-lived in experience that they no longer seem even to arise.
Thus pure emptiness does not properly represent the absence of form but rather form that is so dynamic (in imaginary terms) that it no longer seems to arise in experience.

20. Ken Wilber's failure to recognise the true complementary nature of holons and heaps arises from his one-directional method of interpreting holons.
Thus from the real (conscious) perspective, a holon is a whole that is part of another whole. However equally from the imaginary (unconscious) perspective, a holon (or rather onhol) is a part, which is whole in the context of other parts. Therefore from this perspective the existence of the heap - in any context - is necessary to recognise the whole status of the holon, just as the existence of the holon is necessary to recognise the part aspect of the heap i.e. as a collection of individual parts.
Thus when seen in this sense, holons and heaps are truly interdependent.
So Ken's interpretation of holons and heaps (whereby heaps are treated as a somewhat incoherent aggregate of pre-defined holons) is remarkably reduced and fragmented in nature (and quite inconsistent with an authentic holistic perspective on nature).

21. In Jungian terms the real (conscious) interpretation of the holon relates to the S (sense) function, whereas the imaginary (unconscious) interpretation relates to the N (intuitive) function.
Now Ken Wilber certainly possesses both remarkable S (sense) and N (intuition) capabilities. However the key problem is that there remains a substantial divide as between these two capacities in his work with the S largely confined to linear (asymmetrical) analysis of development structures and the N in turn largely confined to nondual recognition of higher spiritual states.
Thus from one perspective - as for example in his fragmented treatment of holons and heaps - he comes across as an extreme S. From an alternative perspective in his insistence on the absolute nondual nature of Spirit, he comes across as an extreme N.
However just as S and N capabilities necessarily become increasingly interdependent in the integral personality, likewise the interpretation of states and structures (and structures and states) become increasingly interdependent in an integral interpretation.

22. The conventional scientific understanding of dimensions is asymmetrical (i.e. three of space and one in time). Likewise it does not distinguish as between exterior and interior and real (conscious) and imaginary (unconscious) experience.
So an integral interpretation requires incorporating these aspects while maintaining equal balance as between space and time.
The essence of the spiritual nondual experience of reality is that it takes place in the continual present. This requires that the phenomenal positing of phenomena in space and time be negated successfully in dynamic terms so that one remains in the continual present. We then have the relative experience of space and time (where any rigid attachment is negated or cancelled out) sustained by absolute nondual experience of the present moment.
In an Integral 1 interpretation, both space and time have real positive and negative directions (which are purely relative with respect to each other).
In an Integral 2 interpretation both space and time now have real and imaginary directions both positive and negative (again in a purely relative sense).
This would seem to imply that we have now eight dimensions (four of space and four of time).
However the remarkable dynamic fact is that both space and time in integral terms are fully interchangeable with real space = imaginary time, and real time = imaginary space.
What this means in effect that we are enabled to switch in experience between conscious notions of space and time through the hidden presence of their imaginary counterparts.
Clearly when the unconscious is not properly developed, there is limited opportunity for dynamic switching so that rather rigid - merely phenomenal - real notions of dimensions remain.

References

Introduction to Integral Theory and Practice: IOS Basic and AQAL Map - available for download to subscribers on Integral Naked web site
http://integralnaked.org